(2) Since \({M}_B={{L}^2_B}^{\prime \prime }\)  it follows that \({M}_B\)  is the left von Neumann algebra generated by the Hilbert algebra \({L}^{1,2}_{B}\)  (see the discussion and the references at the end of Section REF ). Then \({L}^2_B\)  is a two-sided ideal in \({M}_B\)  in view of [1]}.
